2.1. Smart phone selection/evaluation

Over years, technology and style of mobile phones such as technical features, color, design, size etc. have changed significantly. At the beginning, mobile phones focused were largely limited with voice calling functions. Nowadays, cellular phones offer many more services such as camera, messaging (e.g. SMS, MMS, email), internet access, other connection options (e.g. infrared, Bluetooth), music, business and gaming applications. Cellular phones offering such functions in addition to voice calling are generally referred to as “smart phones”²⁴.

In the current telecommunications market, mobile communication is becoming popular, thus gradually increasing consumers’ desire to use smart phones. Many of the products on the shelves also provide hints about the social and financial status of users, their preferences and attitudes. This turns the selection of a smart phone by a consumer into an important decision problem in which the most appropriate handset is selected even though the user is interested in smart phones in a fixed range of price. As user expectations can vary for each consumer, the selection of a smart phone can be seen as a complex MCDM problem. This study proposes a solid mechanism to support users in deciding on the most suitable mobile phone in the marketplace⁵.

Smart phones are becoming an integral component of daily and business life, so that interest in this technology and in available product alternatives in the market is rising. The Worldwide Quarterly Mobile Phone Tracker published by International Data Corporation suggests that manufacturers have shipped more than 330 million units globally in the 1^st quarter of 2015. Accordingly, the Android operating system dominated the market with more than ¾ share in the same period. Samsung, an electronics manufacturer, leads this large market by offering a wide range of smart phones, offering cutting edge as well as low-cost smart phones. In this picture, Turkey has been one of the forerunners in the adoption of mobile communications. Presently, more than ten global smart phone vendors are actively operating in Turkey, each one having a relatively large product line²⁵. As of 2015, Apple, Nokia, Samsung, LG and HTC are the top five in the Turkish smart phone market. Overall, smart phone penetration continues to rise and the number of mobile broadband subscribers (computer and mobile handset) is around 33.9 million, indicating the intensity of mobile telecommunication in Turkey.

As smart phones become powerful mobile tools, Economides and Grousopoulou²⁶ presented students’ considerations about the importance and costs of the features and services of their smart phones. In their study, the opinions of male and female students are investigated with a survey to gain insight about the importance and costs of these gadgets. Another research from Bayraktar, Tatoğlu, Turkyilmaz, Delen and Zaim²⁵ focused on customer satisfaction and loyalty (CS&L) for existing mobile phone brands in the Turkish market. Hu, Lu and Tzeng²⁷ concentrated on smart phone improvement for promoting product value to satisfy customer needs. Rahul and Majhi²⁸ investigated consumer satisfaction and loyalty in the Indian mobile phone market. Mobile phone use behavior and differences in connection with male and female consumers is evaluated in Haverila’s study²⁴, which investigated the relationship between feature preferences and customer satisfaction and repurchase intent among young male users. An interesting research topic is the mobile phone feature preferences. Işıklar and Büyüközkan⁵, Haverila²⁹ and Haverila³⁰ published articles on this topic. For instance, Haverila³⁰ discussed the progressive evolution of specific smart phone feature preferences in Finland among different high school and college students.

2.2. Smart phone evaluation criteria

Selecting criteria is one of the most important dimensions while constructing a decision model. Therefore, criteria are important components that enable alternatives to be compared from a specific point of view. Generally, users get satisfied with a particular product when its properties match with their preferences and expectations. In order to develop an effective decision model, most important product selection criteria by consumers need to be identified.

In order to prepare this study and develop the proposed selection model, a series of recent publications dealing with the smart phone selection problem are reviewed. Moreover, the opinions of smart phone market experts are taken into account to identify selection criteria. Based on Işıklar and Büyüközkan’s⁵ Haverila’s²⁴, Mokhlis and Yaakop’s³¹, Hu, Lu and Tzeng’s²⁷, Hsiao and Chen’s⁴ studies evaluation model criteria are determined and given in Table 1. Existing literature provides various models on smart phone selection research. For instance, Işıklar and Büyüközkan⁵ and Chen³² concentrated on product and user-related criteria. Işıklar and Büyüközkan⁵ introduced product-related criteria which consist of basic requirements, physical specifications and technical features. The user-related criteria was divided into functionality, brand choice and customer excitement. Another research by Chen, Hu, Kuo and Liang³³ identified the brand, price, hardware feature/functionality, basic built-in functions and extended built-in functions as being the most essential 5 selection criteria for purchasing a smart phone. Mokhlis and Yaakop³¹ summarized the selection criteria as innovative features, image, price, personal recommendation, durability and portable aspects, media influence, and post-sales service. More recently, Hu, Lu and Tzeng²⁷ defined three dimensions which are customer equity, product function and mobile convenience as influencing factors for consumer willingness to purchase a smart phone. Hsiao and Chen⁴ came up with three demand dimensions and emphasized their differences. Accordingly, the smart phone handset, subscription to the 2G/3G network and mobile services are the basic dimensions of consumer demand.

The structure of the proposed model make use of the following criteria: Consumers highly regard physical conditions in selecting their phones, including Durability, Battery life, Changeable parts, Dimensions. Physical conditions refer to the design standards of a product and constitute an essential component. These conditions can differ from product to product. Haverila²⁴ underlined the importance of various feature preferences. This preference consideration becomes particularly important as manufacturers have many feature alternatives to add to a phone model among a large set of potential features. A large number of technical features, however, do not necessarily affect consumers towards a positive purchase decision. Embedding too many features might even have a negative impact on consumers if these features can be perceived to be unnecessary or complicated. Therefore, basic conditions appear to be grouped together.

Technical conditions refer to the technological assistance of products and include Memory capacity, Processor speed, Internet connection speed (4G/5G) and Camera specifications. Beside hardware, consumers also value the software on a mobile phone, i.e. Functionality conditions. Operation system easiness and Variety of applications are among the most interesting features³⁵.

Brand/Market conditions involves the subjective and invisible evaluation of the brand based on the brand awareness, the brand ethics and the brand attitude of the customer. Brand/Market conditions include Brand choice, Prestige, Fashionable style/Aesthetics, Personal information and media security, Price and Warranty/Service Availability. In mobile sector, consumers are facing higher security and privacy risks because of the data transaction in a wireless environment.

2.3. IF-TOPSIS with GDM environment

MCDM is one of the popular methods to deal with complicated problems that exhibit high uncertainty, clashing objectives, various interests and multiple perspectives^32–34. Besides, MCDM methods are effective in decision making, weighting and selecting the most appropriate alternatives^17–18. A number of researchers have used different MCDM techniques in the fields of information, mobile communications, music business and gaming applications^35,36. For instance; Işıklar and Büyüközkan⁵ developed a MCDM technique where they evaluated different mobile phone options with respect to users’ preferences by applying AHP and TOPSIS. Chen, Hu, Kuo and Liang³³ proposed recommendation systems for online mobile phone stores by using an AHP-based mechanism. Hu, Lu and Tzeng²⁷ proposed a hybrid MCDM model for promoting a smart phone’s product value by using DEMATEL, ANP and VIKOR. Apart from these MCDM techniques, this study utilized Intuitionistic Fuzzy Sets (IFS) introduced by Atanassov^{16, 37}. IFS can be seen as the extension of fuzzy sets which was originally proposed by Zadeh³⁸. IFS can be characterized by the components of a membership function, a non-membership function and a margin for hesitation. It is flexible for dealing with uncertainty, whereas fuzzy sets are characterized by only their membership function¹⁶. When the individual evaluations of DMs in a GDM under uncertainty are concerned, it must be taken into account that not all DMs have the same level of knowledge, background and experience. They can differ in terms of skills, personality or area of research^39,40. Therefore, uncertainty is natural for DMs when providing their preferences, exhibiting characteristics of affirmation, negation, and hesitation to some extent^41,42. IFS can in such cases be a helpful tool with its flexibility and robustness^17,18. IFS can also take the degree of hesitation into account and can deal with any error or lack of knowledge in defining the membership function. Beside these advantages, IFS can also cope with uncertain and vague objects. As such, it provides researchers and DMs with a powerful tool for expressing data under different fuzzy environments^17,19.

Integration with GDM can be exemplified in Xu and Liao’s research¹⁹. It is stated that triangular fuzzy numbers, trapezoidal fuzzy numbers and interval-valued fuzzy numbers can only be used to depict the fuzziness of agreement but cannot reflect the disagreement among DMs. However, in real life recognition of human beings’ disagreement is a common expression for effectively describing and communicating opinions. Hence IFS copes with such situations and aggregates experts’ opinions for a collective decision in GDM problems⁴³.

A key approach in GDM problems is to include many alternatives and DMs from different or same disciplines to find the most suitable solution among a set of available alternatives and attempt to reach a collective decision^44,45. Aggregation of expert opinions plays a central role in reaching a collective decision with an evaluation process^46,47. For this purpose, the technique called intuitionistic fuzzy weighted averaging (IFWA) operator is proposed by Xu⁴⁰. IFWA can be used for merging the opinions of each individual DM together, where the aggregated result is used for assigning a value to the importance of selection criteria and available alternatives^12,43. Besides, Xu and Yager⁴⁸ introduced additional intuitionistic fuzzy geometric aggregation operators, such as intuitionistic fuzzy weighted geometric averaging operator (IFWGA), intuitionistic fuzzy ordered weighted geometric averaging operator (IFOWGA) and intuitionistic fuzzy hybrid geometric averaging operator (IFHGA). Xu⁴⁰ presented some other intuitionistic fuzzy aggregation operators, such as intuitionistic fuzzy ordered weighted averaging (IFOWA) operator and the intuitionistic fuzzy hybrid averaging (IFHA) operator. He, Chen, Zhou, Liu and Tao⁴⁹ developed another operator, intuitionistic fuzzy geometric interaction averaging (IFGIA) operator.

Following these valuable developments in literature and other important contributions, researchers recently started to investigate IFS in MCDM. As one of best known MCDM methods, TOPSIS technique was initially proposed by Chen and Hwang⁵⁰. The developed TOPSIS method proposed by Hwang and Yoon¹³ incorporates a simple computation process, systematic procedure, and a solid logic that considers the rationale of people’s choices. In this paper, TOPSIS is utilized as a ranking technique based on its rational logic and understandability¹⁴. Although TOPSIS is very popular to solve MCDM problems, this approach also has some weaknesses. A better approach may be to use IFS rather than fuzzy sets, where criteria ratings and weights are found with intuitionistic numbers. Recently, many researchers extended IF-TOPSIS for MCDM. These methodologies have been applied satisfactorily to different research areas for evaluation purposes and these are summarized in Table 2.

As seen in Table 2, IF-TOPSIS is integrated with different techniques in various fields of application. On the other hand, IF-TOPSIS is utilized with interval valued approach which is different from Boran’s^12–14 proposed approach. To authors’ best knowledge; there exists no publication in which IF-TOPSIS is used for the smart phone selection problem. Therefore, this paper contributes to the literature by addressing this research gap and demonstrating the applicability of the proposed method with a case study.

3.1. Preliminaries

The methodology explained in this section first presents the basic definitions and notations of IFS, most of which are taken from Atanassov’s study.³⁷ In a finite set of X, IFS S can be stated as:

Here, μ_S(x), ν_(S)(x):X→ [0,1] is the membership function and the non membership function respectively, so that,

In IFS, there is another parameter π(x), called the “hesitation degree” that checks if x belongs to S,

Here, for every x ∈ X:

As π_S(x) becomes smaller, the certainty of the knowledge about x becomes higher. As π_S(x) gets higher, then the knowledge about x becomes less certain. In this case, when μ_S(x)= 1 - ν_S(x) for each and every element of the universe, the concept of ordinary fuzzy set is recovered. Defining M and N as two IFSs that belong to the set of X, then the multiplication operator can be defined as the following.³⁹

3.2. IF-TOPSIS

The general view of GDM based approach of IF-TOPSIS is given in “Fig. 1”. The methodology of IF-TOPSIS is adapted from Boran’s studies^12–14. The summary view of this framework starts with identifying evaluation criteria and alternatives using experts’ opinions and a detailed literature review is required to search and collect information. Next, a committee of experts is necessary to provide group decision. According to the group qualifications, different weights are determined for each DM. Then a comparison scale to weight criteria set and rate alternatives are selected. In the next phase an aggregated intuitionistic fuzzy decision matrix is constructed based on DMs assessments. Then criteria weights based on DMs assessments is obtained. The framework concludes with focusing on the selection process. Here, TOPSIS is adapted to the system for ranking available alternatives of smart phones in a decreasing order using their relative closeness coefficient. The steps of the IF-TOPSIS are explained briefly as follows:

Fig. 1.

Schematic diagram of IF-TOPSIS

Define A = {A₁,A₂,…,A_m} as a set of alternatives and X = {X₁,X,…,X_n} as a set of criteria.

• Step 1: Identify the evaluation criteria and alternatives for the smart phone selection problem. The objective is to find out the most appropriate smart phone alternatives among the others.

• Step 2: Find the weights of DMs’ evaluations. Here, the decision committee consists of three DMs. The importance degrees of each of the DM evaluations are processed as linguistic terms expressed in IFS. Assume that D_k = [μ_k, ν_k, π_k] is an intuitionistic fuzzy number with the rating of k^th DM. Accordingly, the weight of the k^th DM can be obtained as:

  (5)   λk=(μk+πk(μkμk+νk))∑k=1l(μk+πk(μkμk+νk))  and∑k=1lλk=1

• Step 3: Select a comparison scale to weight criteria set and rate alternatives. Linguistic label sets with their respective IFS are given in Table 3 and 4 as follows:

  Linguistic term	IFS
  Very Important (VI)	[0.9; 0]
  Important (I)	[0.8; 0.1]
  Moderately Important (MI)	[0.7;0.2]
  Medium (M)	[0.5;0.5]
  Unimportant (U)	[0.3;0.5]
  Very Unimportant (VU)	[0.2;0.7]

  Table 3.

  Linguistic terms for rating criteria

  Linguistic terms	IFS
  Extremely Good (EG)	[1,0, 0]
  Very Good (VG)	[0.75,0.1,0.15]
  Good (G)	[0.6,0.25,0.15]
  Moderately Good (MG)	[0.5,0.4,0.1]
  Medium (M)	[0.5,0.5,0]
  Moderately Bad (MB)	[0.4,0.5,0.1]
  Bad	[0.25,0.6,0.15]
  Very Bad (VB)	[0.1,0.75,0.15]
  Very very Bad (VVB)	[0,0.9,0.1]

  Table 4.

  Linguistic terms for rating alternatives

• Step 4: Form an aggregated intuitionistic fuzzy decision matrix that is based on DMs’ assessments. Assume that R(k)=(rijk)mxn is an intuitionistic fuzzy decision matrix for each DM. Let λ= (λ₁, λ₂,…, λ₁) be the weight of each DM and ∑k=1lλk=1 , λ_k ∊ [0, 1]. Operator IFWA will then be used to aggregate DMs’ evaluations in order to rate the importance values of decision criteria and available alternatives in the GDM process.

  R = (r_ij)_mxn and r_ij = (μ_Ai(x_j),ν_Ai(x_j),π_Ai(x_j)) i=(1,2,..,m; j=1,2,..,n)

  (6)         rij=IFWAλ(rij(1),rij(2),..,rij(l))            =λ1rij(1)⊕λ2rij(2)⊕λ3rij(3)⊕…⊕λlrij(l)=[1−∏k=1l(1−μij(k))λk,∏k=1l(νij(k))λk,∏k=1l(1−μij(k))λk,−∏k=1l(νij(k))λk]

  Then intuitionistic fuzzy decision matrix can be defined as:

  R=[μA1(x1),νA1(x1),πA1(x1)μA1(x2),νA1(x2),πA1(x2)⋯μA1(xn),νA1(xn),πA1(xn)μA2(x1),νA2(x1),πA2(x1)μA2(x2),νA2(x2),πA2(x2)⋯μA2(xn),νA2(xn),πA2(xn)⋮⋮⋱⋮μAm(x1),νAm(x1),πAm(x1)μAm(x2),νAm(x2),πAm(x2)⋯μAm(xn),νAm(xn),πAm(xn)]R=[r11r12r13⋯r1mr21r22r23⋯r2mr31r32r33⋯r3m⋮⋮⋮⋱⋮rn1rn2rn3⋯rnm]

• Step 5: Construct and evaluate weights of criteria according to DMs’ viewpoints. The importance degrees of each criteria can be shown with “W”. Then, in order to evaluate importance degrees, all individual opinions have to be fused. Here, Wj(k)=[μj(k),νj(k),πj(k)] is defined as IFS that is assigned to criteria X_j by the k^th DM. Their criteria weights are calculated as follows:

  (7)  Wj=IFWAλ(Wj(1),Wj(2),..,Wj(l))       =λ1Wj(1)⊕λ2Wj(2)⊕λ3Wj(3)⊕…⊕λlWj(l)   =[1−∏k=1l(1−μj(k))λk,∏k=1l(νj(k))λk,∏k=1l(1−μj(k))λk,−∏k=1l(νj(k))λk]               W=[W1,W2,W3,..,,Wj]            Wj=[μj,νj,πj]  (j=1,2,..,n).

• Step 6: Establish the aggregated weighted intuitionistic fuzzy decision matrix based on the previously constructed criteria weights (W) and the aggregated intuitionistic fuzzy decision matrix, as shown below:

  (8)   R⊗W={〈x,μAi(x). μw(x),νAi(x)+νw(x)−νAi(x).νw(x)〉I x∈X}

  Next,

  (9)πAiw(x)=1−νAi(x)−νw(x)−μAi(x). μw(x)+νAi(x). νw(x)

  Finally, the aggregated weighted intuitionistic fuzzy decision matrix is found as:

  R′=[r′11r′12r′13⋯r′1mr′21r′22r′23⋯r′2mr′31r′32r′33⋯r′3m⋮⋮⋮⋱⋮r′n1r′n2r′n3⋯r′nm]

  r′ij=(μij′,νij′,πij′)=(μAiW(xj),νAiW(xj),πAiW(xj)) is an element of the aggregated weighted intuitionistic fuzzy decision matrix, as found above.

• Step 7: Integrate IF-TOPSIS to aggregated weighted intuitionistic fuzzy decision matrix. Following that, calculate the distances from positive and negative ideal points. Assume that J₁ represents the benefit criteria and J₂ represents the cost criteria. Here, A⁺ is defined as the intuitionistic fuzzy positive-ideal solution and A^- is defined as the intuitionistic fuzzy negative-ideal solution, as shown below:

  (10)A+=((μA+W(xj),νA+W(xj))  andA−=((μA−W(xj),νA−W(xj))     where,
  (11)μA+W(xj)=((maxi μAi.W(xj)|j∊J1),(mini μAi.W(xj)|j∊J2))
  (12)νA+W(xj)=((mini νAi.W(xj)|j∊J1),(maxi νAi.W(xj)|j∊J2))
  (13)μA−W(xj)=((mini μAi.W(xj)|j∊J1),(maxi μAi.W(xj)|j∊J2))
  (14)νA−W(xj)=((maxi νAi.W(xj)|j∊J1),(mini νAi.W(xj)|j∊J2))

• Step 8: Calculate the separation measures of the intuitionistic fuzzy sets of the available alternatives. The distance of each alternative from the positive and negative ideal points are computed as follows:

  (15)S+=12n∑j=1n[(μAi.W(xj)−μA+W(xj))2+(νAi.W(xj)−νA+W(xj))2+(πAiW(xj)−πA+W(xj))2]
  (16)S−=12n∑j=1n[(μAi.W(xj)−μA−W(xj))2+(νAi.W(xj)−νA−W(xj))2+(πAiW(xj)−πA+W(xj))2]

• Step 9: Find the relative closeness coefficient (CC_i) for the intuitionistic ideal solution. For an alternative A_i, its CC_i with respect to A⁺ can be found as:

  (17)CCi=Si−Si−+Si+

• Step 10: Rank the alternatives in decreasing order of CC_i values. At this step, the alternative which has the maximum CC_i is selected.

5.1. Comparison with Chen’s fuzzy TOPSIS

A comparative analysis is carried out to investigate the consistency of the rank and weight of the alternatives selection. The work that makes use of IF-TOPSIS is tested and compared with the results of Chen’s⁶⁰ fuzzy TOPSIS method. Table 12 presents the ranking of the alternatives according to their performance indices. According to the overall result of Fuzzy TOPSIS, A2 is the best alternative followed by A1 and A3 (A2>A1>A3). The ranking of the alternatives changes very small comparing to the results of utilized IF-TOPSIS method (A1>A2>A3).

From this research, the ranking orders are slightly inconsistent with Fuzzy TOPSIS due to use of different preference scales of the DMs. However, the utilized preference scale with hesitation degree is based on the IFS notation. As a summary, it can be seen that the differences in the values may come from the intuitionistic evaluations of the utilized method, providing more flexible and informative definitions of fuzzy sets.

5.2. Sensitivity analysis

A sensitivity analysis is performed in order to determine whether the final solution is robust to changes of the weights of a specific expert. Considering that the priorities are remarkably dependent on subjective judgments of the DMs, the stability of the final ranking under different weights should be checked out. With regard to this purpose, it is better to execute a sensitivity analysis based on a set of cases that reflect different views on the relative importance of the determinants. By altering the weights and providing some insights into the results are the main idea of the analysis. Initially, DM1 has “medium”, DM2 and DM3 have “very important” linguistic importance weights. This is the current situation and named as CASE 1. In the sensitivity analysis (see Table 13) the importance weights are changed to “very important” for DM1, “medium” for DM2 and “important” for DM3 - named as CASE 2. In another case, named as CASE3, weights are changed to “important” for DM1, “very important” for DM2 and “medium” for DM3. The results of cases are presented in Tables 14 and 15.

It is seen that as no significant changes occur in the most important alternative, the results are not sensitive to the importance weights of DMs. In other words, the sensitivity analysis shows that the changes in the DMs’ weights do not cause any change in the ranking of the considered smart phone alternatives. This means that our decision is robust against possible changes in DMs’ weights.